On Two Group Functors Extending Schur Multipliers

Abstract

Liedtke has introduced group functors K and $\tilde{K}$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work, we relate K and $\tilde{K}$ to a group functor τ arising in the construction of the non-abelian exterior square of a group. In contrast to $\tilde{K}$, there exist efficient algorithms for constructing τ, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $\tau (G)$, and when $\tau (G)$ and $\tilde{K}(G,3)$ are isomorphic.

Keywords:

• Finite groups
• Schur multiplier
• non-abelian exterior square

2010 Mathematics Subject Classification:

• 20J06
• 14J29

Acknowledgments

Both authors thank the referees for the thorough reading and for providing many details that made some of our results stronger.

Declaration of interest

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported through the programme “Research in Pairs” by the Mathematisches Forschungsinstitut Oberwolfach in 2018; the authors thank the MFO for the great hospitality. Moravec acknowledges the financial support from the Slovenian Research Agency (research core funding no. P1-0222, and projects no. J1-8132, J1-7256 and N1-0061). Dietrich’s research is supported by an Australian Research Council grant, identifier DP190100317.